Mister Exam

Lang:

Other calculators

How to use it?
Integral of d{x}:
Integral of e^(-x^3)
Integral of (x^2)
Integral of |x|+1dx
Integral of dx/(3*x-7)
Identical expressions
(x^ two)*exp(x/ two)
(x squared ) multiply by exponent of (x divide by 2)
(x to the power of two) multiply by exponent of (x divide by two)
(x2)*exp(x/2)
x2*expx/2
(x²)*exp(x/2)
(x to the power of 2)*exp(x/2)
(x^2)exp(x/2)
(x2)exp(x/2)
x2expx/2
x^2expx/2
(x^2)*exp(x divide by 2)
(x^2)*exp(x/2)dx

Integral of (x^2)*exp(x/2) dx

The solution

You have entered [src]

 oo         
  /         
 |          
 |      x   
 |      -   
 |   2  2   
 |  x *e  dx
 |          
/           
0

$$\int\limits_{0}^{\infty} x^{2} e^{\frac{x}{2}}\, dx$$

Integral(x^2*exp(x/2), (x, 0, oo))

Detail solution

Use integration by parts:

$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$

Let $u{\left(x \right)} = x^{2}$ and let $\operatorname{dv}{\left(x \right)} = e^{\frac{x}{2}}$ .

Then $\operatorname{du}{\left(x \right)} = 2 x$ .

To find $v{\left(x \right)}$ :
1. Let $u = \frac{x}{2}$ .
  
  Then let $du = \frac{dx}{2}$ and substitute $2 du$ :
  
  $\int 2 e^{u}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\text{False}$
    1. The integral of the exponential function is itself.
      
      $\int e^{u}\, du = e^{u}$
    So, the result is: $2 e^{u}$
  Now substitute $u$ back in:
  
  $2 e^{\frac{x}{2}}$
Now evaluate the sub-integral.
Use integration by parts:

$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$

Let $u{\left(x \right)} = 4 x$ and let $\operatorname{dv}{\left(x \right)} = e^{\frac{x}{2}}$ .

Then $\operatorname{du}{\left(x \right)} = 4$ .

To find $v{\left(x \right)}$ :
1. Let $u = \frac{x}{2}$ .
  
  Then let $du = \frac{dx}{2}$ and substitute $2 du$ :
  
  $\int 2 e^{u}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\text{False}$
    1. The integral of the exponential function is itself.
      
      $\int e^{u}\, du = e^{u}$
    So, the result is: $2 e^{u}$
  Now substitute $u$ back in:
  
  $2 e^{\frac{x}{2}}$
Now evaluate the sub-integral.
The integral of a constant times a function is the constant times the integral of the function:

$\int 8 e^{\frac{x}{2}}\, dx = 8 \int e^{\frac{x}{2}}\, dx$
1. Let $u = \frac{x}{2}$ .
  
  Then let $du = \frac{dx}{2}$ and substitute $2 du$ :
  
  $\int 2 e^{u}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\text{False}$
    1. The integral of the exponential function is itself.
      
      $\int e^{u}\, du = e^{u}$
    So, the result is: $2 e^{u}$
  Now substitute $u$ back in:
  
  $2 e^{\frac{x}{2}}$
So, the result is: $16 e^{\frac{x}{2}}$
Now simplify:

$2 \left(x^{2} - 4 x + 8\right) e^{\frac{x}{2}}$
Add the constant of integration:

$2 \left(x^{2} - 4 x + 8\right) e^{\frac{x}{2}}+ \mathrm{constant}$

The answer is:

$2 \left(x^{2} - 4 x + 8\right) e^{\frac{x}{2}}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                       
 |                                        
 |     x              x        x         x
 |     -              -        -         -
 |  2  2              2        2      2  2
 | x *e  dx = C + 16*e  - 8*x*e  + 2*x *e 
 |                                        
/

$$\int x^{2} e^{\frac{x}{2}}\, dx = C + 2 x^{2} e^{\frac{x}{2}} - 8 x e^{\frac{x}{2}} + 16 e^{\frac{x}{2}}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

oo

$$\infty$$

oo

$$\infty$$

oo

Use the examples entering the upper and lower limits of integration.

Other calculators

How to use it?

Identical expressions

Integral of (x^2)*exp(x/2) dx

Limits of integration:

The graph:

Piecewise:

The solution